Without drawing any parallels from Jewish belief or liturgy, from the standpoint of a popular understanding of Islam, the next day is understood to begin after the setting of the sun.

Notwithstanding the fact that a 24 hour count can begin at any point of the day, from a Quranic perspective at least, it can be argued that the start of the day is generally understood to commence with dawn (Arabic: Fajr) and not the setting of the sun.

There is some support as can be noted in the brief points below:
 

1. If there are five prayers to the day [1] and if popular Islamic thought is accepted as to when day begins (sunset), the night prayer would then logically be the first prayer of the next day. This appears difficult to accept, especially in light of the following point (2).

2. The mention of 'middle prayer' in verse 2:238 implies a mid-point prayer as part of a complete day. The mid-point prayer would be ‘Asr’ with 2 prayers after it (Maghrib and Isha) and 2 prayers before it (Fajr & Dhuhr). [2]

002:238

“Guard (Arabic: Hafizu) strictly your prayers (Arabic: salawaat), especially the middle (Arabic: wusta) prayer; and stand before God in a devout (frame of mind)” 

3. Verses 91:1-4 seem to imply a day relationship with the brightness which appears with pre-dawn and complete rising of the sun. The ‘nahar’ ends with the onset of the night (layl) but this does not imply that the ‘day’ (yaum) is complete. The complete day arguably includes the night and completed only until the next pre-dawn when the sun's rays light up the sky again.

091:001-4
"By the sun and its brightness and the moon when it follows it. And the day (Arabic: nahar) when it displays it and the night when it covers / enshrouds it"

This is again supported by the following point (4)

4. Verse 11:114 refers to establishing salaat at two ends of the 'nahar' with the additional approach to the night. Therefore, the night prayer is arguably included as part of the complete 'day-cycle' and does not form part of the next day as commonly accepted.

011:114 (part)

“And establish regular prayers at the two ends (Arabic: Salata Tarafa) of the day (Arabic: nahar) and at the approaches of the night (Arabic: wa-zulafan mina al-layl)..."

5. In verse 69:7, Aad was destroyed by a fierce roaring wind which was unleashed on them for '7 nights and 8 days (ayyam)' in succession (husum-uninterrupted).

069:007
"Which He imposed upon them (for) seven nights and eight days (in) succession / uninterrupted (Arabic: husum)..."

If the day (yaum) were to commence at night (after sunset) then the 8th day would commence on the 8th night and continue into the 8th night. In other words, the 8th day cannot commence without entering into the 8th night.
However, if the day (yaum) were to commence at dawn (fajr) then the 8th day would commence with the 8th dawn but with only 7 nights preceding it (as in verse 69:7) and the 8th day would not enter the 8th night immediately.
Hence, the latter scenario appears to be a better representation of the Quranic data which arguably, better intimates the commencement of ‘a day’ with dawn (fajr).

Please see the following illustration below:

6. Verse 97:5 of Surah al-Qadr implies that the new day begins with the ‘rising of the dawn’ (salamun hiya hatta matla-il-fajr - Peace until the emergence / break of the dawn). As verses 97:1-4 are referring to a 'night' of power, the demarcation of the 'emergence / breaking of the dawn' and its strong implication with the initiation of the new day remains noteworthy.

7. Fasting only commences from ‘fajr' - dawn (2:187) and ends at layl (night) not sunset [3]. If the next day was to commence after sunset, then fasting would be logically understood to start in the midst of the day cycle which would make the term ‘night of the fast’ somewhat problematic.

For example, in verse 2:187, permission is granted to approach one’s wives. If the day allegedly commences before the first fast of Ramadan, then this night does not appear to cogently reconcile with the term ‘night of the fasting’ especially in light of the permission that has been granted for conjugal relationships if the actual fasting period hasn’t even commenced. However, once fasting has commenced, ‘the night of the fasts’ and permission for conjugal relationships thereon can easily be reconciled with the first night after the fasting has commenced.

8. Verse 2:184 asks believers to fast a certain number of days (ayyaman) implying that the day is expected to commence when the fast commences which is at dawn (fajr) - 2:187.

9. Notwithstanding that verse 25:47 presents possible analogies of sleep with death and the awakening with resurrection / the act of reviving, the day (nahar) appears to be twinned with a new start, a new period, a rising or a resurrection (nushur). Hence from this verse, the night could be argued as a completion of a period and the day (once awakened) a new start, a new beginning.

 025:047
"And He is the One who made for you the night (Arabic: layl) a covering and the sleep a rest and made the day (Arabic: nahar) a resurrection (Arabic: nushur)"
 

It is important to remember that albeit that the sighting of the crescent of the moon in the evening shows the start of the next month, this does not mean that its sighting ushers the start of a ‘new day’. There remains a distinction, certainly as posited above, from a Quran’s perspective.

From a classical Arabic perspective, it also appears that there is support for the above perspective that the new day initiates with dawn and that a day can only start when the sun is above the earth.

Source: Edward Lanes Lexicon    [4]

 

FINAL THOUGHTS

Contrary to popular assertions and beliefs, there appears to be no explicit support from the Quran that the next ‘day’ commences at 'sunset'. Rather, there appears to be stronger evidence from a Quran's perspective to understand a day to start from dawn and end with sunset. This is also supported by classical Arabic sources.